Are uncorrelated normal random variables necessarily independent?

Professor Rosenthal nicely explained this questions by giving two clear examples. The golden quote is "What is true is that if the random variable pair (X,Y) follows the bivariate normal distribution, and Cov(X,Y) = 0, then X and Y must be independent. But what is not true is that if each of X and Y is normally distributed, and Cov(X,Y) = 0, then X and Y must be independent".